CBSE Class 12 Maths 2025 Question Paper 65/6/3 with Solutions

If ${\tan }^{-1}(x^2-y^2)=a$, where ‘a’ is a constant, then $\frac{dy}{dx}$ is :

1. (A)$\frac{x}{y}$
2. (B)$-\frac{x}{y}$
3. (C)$\frac{a}{x}$
4. (D)$\frac{a}{y}$

If $A=\left[\begin{array}{ccc}0& 0& -5\\ 0& 3& 0\\ 4.3& 0& 0\end{array}\right]$, then A is a :

1. (A)skew-symmetric matrix
2. (B)scalar matrix
3. (C)diagonal matrix
4. (D)square matrix

The graph shown below depicts :

1. (A)$y={\sec }^{-1}x$
2. (B)$y=\sec x$
3. (C)$y={\text{cosec}}^{-1}x$
4. (D)$y=\text{cosec}x$

$\left[{\sec }^{-1}(-\sqrt{2})-{\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)\right]$ is equal to :

1. (A)$\frac{11\pi }{12}$
2. (B)$\frac{5\pi }{12}$
3. (C)$-\frac{5\pi }{12}$
4. (D)$\frac{7\pi }{12}$

Let both $A{B}^{\prime }$ and $B^{\prime }A$ be defined for matrices A and B. If order of A is $n\times m$, then the order of B is :

1. (A)$n\times n$
2. (B)$n\times m$
3. (C)$m\times m$
4. (D)$m\times n$

Sum of two skew-symmetric matrices of same order is always a/an :

1. (A)skew-symmetric matrix
2. (B)symmetric matrix
3. (C)null matrix
4. (D)identity matrix

If $y=a\cos (\log x)+b\sin (\log x)$, then $x^2{y}_2+x{y}_1$ is :

1. (A)$\cot (\log x)$
2. (B)y
3. (C)$-y$
4. (D)$\tan (\log x)$

If $f(x)=\{\begin{array}{ll}\frac{\log (1+ax)+\log (1-bx)}{x},& \text{for }x\ne 0\\ k,& \text{for }x=0\end{array}$ is continuous at $x=0$, then the value of k is :

1. (A)a
2. (B)$a+b$
3. (C)$a-b$
4. (D)b

$f(x)=x^x$ has a critical point at :

1. (A)$x=e$
2. (B)$x=e^{-1}$
3. (C)$x=0$
4. (D)$x=1$

The solution for the differential equation $\log \left(\frac{dy}{dx}\right)=3x+4y$ is :

1. (A)$3e^{4y}+4e^{-3x}+C=0$
2. (B)$e^{3x+4y}+C=0$
3. (C)$3e^{-3y}+4e^{4x}+12C=0$
4. (D)$3e^{-4y}+4e^{3x}+12C=0$

For a Linear Programming Problem (LPP), the given objective function is $Z=x+2y$. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph.
(Note : The figure is not to scale)
$P\equiv \left(\frac{3}{13},\frac{24}{13}\right)$, $Q\equiv \left(\frac{3}{2},\frac{15}{4}\right)$, $R\equiv \left(\frac{7}{2},\frac{3}{4}\right)$, $S\equiv \left(\frac{18}{7},\frac{2}{7}\right)$
Which of the following statements is correct ?

1. (A)Z is minimum at $S\left(\frac{18}{7},\frac{2}{7}\right)$
2. (B)Z is maximum at $R\left(\frac{7}{2},\frac{3}{4}\right)$
3. (C)(Value of Z at P) $>$ (Value of Z at Q)
4. (D)(Value of Z at Q) $<$ (Value of Z at R)

The order and degree of the differential equation ${\left[\frac{d^{2}y}{d{x}^2}-1\right]}^2=\frac{dy}{dx}$ are, respectively :

1. (A)2, 2
2. (B)2, not defined
3. (C)1, 2
4. (D)1, not defined

Let $f^{\prime }(x)=3(x^2+2x)-\frac{4}{x^3}+5$, $f(1)=0$. Then, f(x) is :

1. (A)$x^3+3x^2+\frac{2}{x^2}+5x+11$
2. (B)$x^3+3x^2+\frac{2}{x^2}+5x-11$
3. (C)$x^3+3x^2-\frac{2}{x^2}+5x-11$
4. (D)$x^3-3x^2-\frac{2}{x^2}+5x-11$

In a Linear Programming Problem (LPP), the objective function $Z=2x+5y$ is to be maximised under the following constraints :
$x+y\le 4$, $3x+3y\ge 18$, $x,y\ge 0$
Study the graph and select the correct option.
(Note : The figure is not to scale)
The solution of the given LPP :

1. (A)lies in the shaded unbounded region.
2. (B)lies in $△$ AOB.
3. (C)does not exist.
4. (D)lies in the combined region of $△$ AOB and unbounded shaded region.

The area of the region bounded by the curve $y^2=x$ between $x=0$ and $x=1$ is :

1. (A)$\frac{3}{2}$ sq units
2. (B)$\frac{2}{3}$ sq units
3. (C)3 sq units
4. (D)$\frac{4}{3}$ sq units

$\int \frac{x+5}{(x+6{)}^2}{e}^{x}dx$ is equal to :

1. (A)$\log (x+6)+C$
2. (B)$e^x+C$
3. (C)$\frac{e^x}{x+6}+C$
4. (D)$\frac{-1}{(x+6{)}^2}+C$

Let $|\vec{a}|=5$ and $-2\le \lambda \le 1$. Then, the range of $|\lambda \vec{a}|$ is :

1. (A)$[5,10]$
2. (B)$[-2,5]$
3. (C)$[-2,1]$
4. (D)$[-10,5]$

A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up is 0.10, member B does not turn up is 0.20 and member C does not turn up is 0.05. The probability of the meeting being cancelled is :

1. (A)0.35
2. (B)0.316
3. (C)0.001
4. (D)0.65

Assertion (A) : If $|\vec{a}\times \vec{b}{|}^2+|\vec{a}\cdot \vec{b}{|}^2=256$ and $|\vec{b}|=8$, then $|\vec{a}|=2$.
Reason (R) : ${\sin }^2\theta +{\cos }^2\theta =1$ and $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$ and $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Assertion (A) : Let $f(x)=e^x$ and $g(x)=\log x$. Then $(f+g)x=e^x+\log x$ where domain of $(f+g)$ is R.
Reason (R) : $\text{Dom}(f+g)=\text{Dom}(f)\cap \text{Dom}(g)$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that BC = 3BA.

Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.

Find the domain of $f(x)={\sin }^{-1}(-x^2)$.

Find the interval in which $f(x)=x+\frac{1}{x}$ is always increasing, $x\ne 0$.

Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.

If $(x{)}^y=(y{)}^x$, then find $\frac{dy}{dx}$.

Find the angle at which the given lines are inclined to each other :
$l_1:\frac{x-5}{2}=\frac{y+3}{1}=\frac{z-1}{-3}$
$l_2:\frac{x}{3}=\frac{y-1}{2}=\frac{z+5}{-1}$

Find the value of x, if $\left[\begin{array}{ccc}1& x& 1\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\\ 2& 5& 1\\ 15& 3& 2\end{array}\right]\left[\begin{array}{c}1\\ 2\\ x\end{array}\right]=O$.

Find the distance of the point P(2, 4, $-1$) from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

Let the position vectors of the points A, B and C be $3\hat{i}-\hat{j}-2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+5\hat{j}+3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

Consider the Linear Programming Problem, where the objective function $Z=(x+4y)$ needs to be minimized subject to constraints
$2x+y\ge 1000$
$x+2y\ge 800$
$x,y\ge 0$.
Draw a neat graph of the feasible region and find the minimum value of Z.

A student wants to pair up natural numbers in such a way that they satisfy the equation $2x+y=41$, $x,y\in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.

Show that the function $f:N\to N$, where N is a set of natural numbers, given by $f(n)=\{\begin{array}{ll}n-1,& \text{if }n\text{ is even}\\ n+1,& \text{if }n\text{ is odd}\end{array}$ is a bijection.

Differentiate $y={\sin }^{-1}(3x-4x^3)$ w.r.t. x, if $x\in \left[-\frac{1}{2},\frac{1}{2}\right]$.

Differentiate $y={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ with respect to x, when $x\in (0,1)$.

Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly by from bag I is transferred to bag II and then a ball is drawn randomly from bag II. Find the probability that the ball drawn is white.

Solve the differential equation : $x^{2}ydx-(x^3+y^3)dy=0$.

Solve the differential equation $(1+x^2)\frac{dy}{dx}+2xy-4x^2=0$ subject to initial condition $y(0)=0$.

Using integration, find the area of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ bounded between the lines $x=-\frac{a}{2}$ to $x=\frac{a}{2}$.

Find :
$\int \frac{x^2+1}{(x-1{)}^2(x+3)}dx$

Evaluate :
${\int }_0^{\pi /2}\frac{x}{\sin x+\cos x}dx$

Show that the line passing through the points A (0, $-1$, $-1$) and B (4, 5, 1) intersects the line joining points C (3, 9, 4) and D ($-4$, 4, 4).